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Advice for specialized undergraduates August 21, 2009

Posted by David Speyer in math life, Uncategorized.
21 comments

Timothy Burke, a professor of History at Swarthmore, writes

[L]et’s just say that you’re a prospective undergraduate who wants to study one subject more than any other… [H]ere’s how I think a prospective who self-identifies as highly interested in one topic or subject ought to work through the questions involved.

First, are you sure that you’re really that interested in a single topic or issue, so sure that you want to make that a primary axis of your decision about where to go to college? Why are you that sure? Do you just like the topic or are you thinking already of a profession narrowly based on it? Are you sure based on an understanding of what a likely undergraduate-level curriculum around that topic looks like, or based on what you know about it from your high school experience? Are you making that choice with a wider awareness of the subjects that even a small college will offer to you that virtually no high school curriculum can focus on?

Second, are you SURE? Really? Then you’re a really unusual applicant. Most of what prospectives think they’re interested in is not the same as what those subjects turn out to be, and most of their interests are based on a very incomplete understanding of the range of academic subjects even within a particular discipline.

Third, if you’re really that kind of unusual person, absolutely certain that your first, second and last priority is to comprehensively study a single subject area while you’re an undergraduate and that this priority is unlikely to change, then: a) don’t apply to any small undergraduate institution; b) pick a place with as few general education requirements as possible; c) find a program in your preferred subject at a large institution that is stuffed to the gills with faculty and courses and make sure undergraduates with a dedicated interest get access to the most prestigious or high-powered faculty in your area of subject interest. The relative difference between one small college and the next doesn’t really matter to you if you’re that driven, because in either case, they’re going to have a relative paucity of resources in comparison to a large institution. You don’t really care about any of the other resources at an institution if you’re that focused: just your area of study and whatever direct supporting skill areas you need (say, language or quantitative training). An undergraduate applicant who is this specifically focused is really more like a proto-graduate student, and should use selection rules much closer to what a graduate student might employ.

This sort of student may be extremely unusual in history, but I would say that they are only somewhat unusual in mathematics. By time I got to Harvard, I knew I wanted to be a mathematician and was struggling to learn as much as I could, as fast as I could. I was not at all the most sophisticated student in my year, and I think that the top undergraduates and high school students are noticeably better now then we were then.

So, I thought it might be interesting to see which of our readers identify with Burke’s hypothetical student, and what they think of his advice.

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A question for the combinatorialists August 21, 2009

Posted by Ben Webster in blegs, combinatorics.
4 comments

One of the points of combinatorics I never really learned is how to play correctly with Mobius functions. I mean, I can state Mobius inversion for an arbitrary poset if you give me a moment, but it all ends up a bit hard to manipulate.

This is particularly frustrating since I know that there are a certain number of people out there in the world who know all these tricks by heart. One of them should make a cheat sheet for all these identities.

So, here’s a question that might be easy to answer, but that I can’t quite muddle my way through. Let P be a ranked poset (in my example, it’s flats of a matroid) with unique minimal element 0 and maximal 1 and Mobius function \mu. Is there any better expression for the sum

\sum_{p\in P} (-1)^{\mathrm{rk}(p)}\mu(0,p)\mu(p,1)?

Algebraic geometry without prime ideals August 6, 2009

Posted by Joel Kamnitzer in Algebraic Geometry, Anton Geraschenko, things I don't understand.
78 comments

The first definition in “Grothendieck-style” algebraic geometry is the affine scheme Spec R for any ring R. This is a topological space whose set of points in the set of prime ideals in R. Then one defines a scheme to be a locally ringed space locally isomorphic to an affine scheme.

The definition of Spec R goes against intuition since it involves prime ideals, not just maximal ideals. Maximal ideals are more natural, since if R = k[x_1, \dots, x_n]/I for some alg closed field k , then the set of maximal ideals of R is in bijection with the vanishing set in the affine space k^n of the ideal I . (Of course one can give a geometric meaning to the prime ideals in terms of subvarieties, but it is less natural.)

However, in Daniel Perrin’s text Algebraic geometry, an introduction, he states/implies that one can define affine schemes just using maximal ideals (at least for finitely-generated k algebras) and still get a good theory of schemes and varieties. Is this true?  If so why don’t we all learn it this way? (One answer to the this latter question could be that some people are interested in non-algebraically closed fields.)

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Bleg: Simplicial Complex Reference August 4, 2009

Posted by David Speyer in Uncategorized.
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Post deleted because the construction it asked about was even dumber than I realized.

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